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Statistics > Methodology

Title:Estimation for general birth-death processes

Abstract: Birth-death processes (BDPs) are continuous-time Markov chains that track the
number of "particles" in a system over time. While widely used in population
biology, genetics and ecology, statistical inference of the instantaneous
particle birth and death rates remains largely limited to restrictive linear
BDPs in which per-particle birth and death rates are constant. Researchers
often observe the number of particles at discrete times, necessitating data
augmentation procedures such as expectation-maximization (EM) to find maximum
likelihood estimates. The E-step in the EM algorithm is available in
closed-form for some linear BDPs, but otherwise previous work has resorted to
approximation or simulation. Remarkably, the E-step conditional expectations
can also be expressed as convolutions of computable transition probabilities
for any general BDP with arbitrary rates. This important observation, along
with a convenient continued fraction representation of the Laplace transforms
of the transition probabilities, allows novel and efficient computation of the
conditional expectations for all BDPs, eliminating the need for approximation
or costly simulation. We use this insight to derive EM algorithms that yield
maximum likelihood estimation for general BDPs characterized by various rate
models, including generalized linear models. We show that our Laplace
convolution technique outperforms competing methods when available and
demonstrate a technique to accelerate EM algorithm convergence. Finally, we
validate our approach using synthetic data and then apply our methods to
estimation of mutation parameters in microsatellite evolution.